The motion estimation algorithm and motion compensation algorithm are core algorithms in the video encoding and decoding technology. The principle of motion estimation is as follows: during the encoding, the currently encoded image block is used as the unit, and a proper matching block is searched in a reference image; a residual is obtained by subtracting the pixel value of the matching block from the pixel value of the image block to remove the redundant information of the image; the residual is converted to remove the redundant information once again, and the entropy encoding is performed on the value obtained after the variation coefficient is quantized; the bit data obtained after the entropy encoding is written to a bit stream; the motion offset is also written to the bit stream after undergoing the encoding, where the motion offset is the difference between the position of the matching block in the reference image and the position of the currently encoded image block in the current image. The offset is also called a motion vector. The principle of motion compensation is as follows: during the decoding, the entropy encoding information is retrieved from the bit data in the bit stream and undergoes the entropy encoding; the residual obtained in the motion estimation is obtained after the entropy encoding information undergoes inverse quantization and inverse conversion, and the motion vector is obtained from the bit stream; a corresponding matching block is obtained in the reference image according to the motion vector value, and the pixel value after the current image block is decoded is equal to the sum of the pixel value of the matching block and the pixel value of the reference image in the residual value. Because the residual between the currently encoded image block and the matching block is written to the bit stream, the redundant information of the currently encoded image block is greatly reduced, which compresses the information written to the bit stream and thus improves the encoding efficiency.
During the motion estimation, to find a more accurate matching block in the reference image, the redundant information may be further reduced. The encoder/decoder generally uses search based on the sub-pixel accuracy. Integer pixel points are original pixel points of an original image or an encoded/decoded image. After multiple integer pixel points are processed, the pixel points obtained between the integer pixel points are called sub-pixel points. That is, the sub-pixel points between two integer pixel points may be called one-half sub-pixel points (hereinafter referred to as ½ sub-pixel points), for example, b, h, and j shown in FIG. 1. The sub-pixel points between two ½ sub-pixel points or between an integer pixel point and a ½ sub-pixel point may be called one-quarter sub-pixel points (hereinafter referred to as ¼ sub-pixel points), for example, a c, d, l, f, n, I, k, e, g, m, and o shown in FIG. 1. The process of obtaining sub-pixel points may be called an interpolation filter process. The search based on the sub-pixel accuracy may obtain a better image matching block, which may reduce the redundant information of the currently encoded image block and improve the encoding efficiency. Thus, the interpolation filter of pixel points determines the effect of the search based on the sub-pixel accuracy. In addition, in the interpolation filter of high-accuracy pixel points, the complexity of the interpolation method is also an important factor in the current video encoding and decoding technology.
Standards related to the video encoding and decoding technology provide interpolation methods based on ¼ sub-pixels, for example, ¼ sub-pixel interpolation in the following solution 1 and adaptive ¼ sub-pixel interpolation in the following solutions 2 and 3.
In the current video compression encoding standards such as H.264/AVC and AVS, interpolation methods based on the ½ sub-pixel accuracy and ¼ sub-pixel accuracy are generally used. The interpolation methods and interpolation coefficients used in the interpolations are fixed. That is, all the image blocks in an image have the same interpolation methods based on the sub-pixel accuracy; and the ½ sub-pixel points and ¼ sub-pixel points at the same position in the image blocks have the same interpolation methods and interpolation coefficients. The following are examples.
Solution 1: As shown in FIG. 1, C3, C4, D3, and D4 are integer pixel points; b, h, and j are ½ sub-pixel points to be interpolated between these four integer pixel points; a, c, d, l, f, n, I, k, e, g, m, and o are ¼ sub-pixel points to be interpolated between these four integer pixel points. The following describes specific interpolation methods for the preceding sub-pixel points.
The interpolations of the ½ sub-pixel points areas follows:b=(C1−5×C2+2×C3+2×C4−5×C5+C6)/32;h=(A3−5×B3+2×C3+2×D3−5×E3+F3)/32;j=(bA−5×bB+20×b+20×bD−5×bE+bF)/32;
bA indicates the ½ sub-pixel point between A3 and A4; bB indicates the ½ sub-pixel point between B3 and B4; the rest may be deduced in the same way.
The interpolations of the ¼ sub-pixel points are as follows:
a=(C3+b)/2; c=(b+C4)/2; d=(C3+h)/2; 1=(h+D3)/2;
f=(b+j)/2; n=(j+bD)/2; i=(h+j)/2; k=(H4+j)/2;
e=(b+h)/2; g=(b+H4)/2; m=(h+bD)/2; o=(bD+H4)/2;
H4 indicates the ½ sub-pixel point between C4 and D4.
The following describes specific interpolation methods of the sub-pixel points in the AVS standard, as shown in FIG. 1.
The interpolations of the ½ sub-pixel points are as follows:b=(−C2+5C3+5C4−C5)/8;h=(−B3+5C3+5D3−E3)/8;j=(−bB+5b+5bD−bE)/8;
The interpolations of the ¼ sub-pixel points are as follows:a=(−C1−2C2+96C3+42C4−7C5)/128;c=(−7C2+42C3+96C4−2C5−C6)/128;i=(−H1−2H2+96h+42H4−7H5)/128;k=(−7H2+42h+96H4−2H5−H6)/128;
H1 indicates the ½ sub-pixel point between C1 and D1; H2 indicates the ½ sub-pixel point between C2 and D2; the rest may be deduced in the same way.d=(−C1−2C2+96C3+42C4−7C5)/128;l=(−7C2+42C3+96C4−2C5−C6)/128;f=(−bA−2bB+96b+42bD−7bE)/128;n=(−7bB−42b+96bD−2bE−bF)/128;
bA indicates the ½ sub-pixel point between A3 and A4; bB indicates the ½ sub-pixel point between B3 and B4; the rest may be deduced in the same way.
e=(C3+j)/2; o=(D4+j)/2;
g=(C4+j)/2; m=(D3+j)/2;
In the current sub-pixel interpolation algorithm, the interpolation coefficients are fixed. As a result, the interpolation method cannot be adapted to the change of the image texture and structure information. In the H.264/AVC standard, only weighted averaging is performed on two adjacent points of the ¼ sub-pixel points, which greatly reduces the interpolation accuracy. In the AVS standard, only weighted averaging is performed on two neighbor points of e, g, m, and o, which greatly reduces the interpolation accuracy. Thus, the search accuracy of sub-pixel points is not high.
Solution 2: In the current VCEG standard, a frame-level adaptive two-dimensional interpolation filter method is available. Details are as follows:
                              a          =                    ⁢                      [                                          C                ⁢                                                                  ⁢                                  1                  ·                                      h                                          1                      ⁢                      a                                                                                  +                              C                ⁢                                                                  ⁢                                  2                  ·                                      h                                          2                      ⁢                      a                                                                                  +                              C                ⁢                                                                  ⁢                                  3                  ·                                      h                                          3                      ⁢                      a                                                                                  +                                                                                            ⁢                                          C                ⁢                                                                  ⁢                                  4                  ·                                      h                                          4                      ⁢                      a                                                                                  +                              C                ⁢                                                                  ⁢                                  5                  ·                                      h                                          5                      ⁢                      a                                                                                  +                              C                ⁢                                                                  ⁢                                  6                  ·                                      h                                          6                      ⁢                      a                                                                                  +                              2                9                                      ]                    >>          10                                                  b          =                    ⁢                      [                                                            (                                                            C                      ⁢                                                                                          ⁢                      1                                        +                                          C                      ⁢                                                                                          ⁢                      6                                                        )                                ·                                  h                                      1                    ⁢                    b                                                              +                                                (                                                            C                      ⁢                                                                                          ⁢                      2                                        +                                          C                      ⁢                                                                                          ⁢                      5                                                        )                                ·                                                                                                          ⁢                                          h                                  2                  ⁢                  b                                            +                                                (                                                            C                      ⁢                                                                                          ⁢                      3                                        +                                          C                      ⁢                                                                                          ⁢                      4                                                        )                                ·                                  h                                      3                    ⁢                    b                                                              +                              2                9                                      ]                    >>          10                                                  c          =                    ⁢                      [                                          C                ⁢                                                                  ⁢                                  1                  ·                                      h                                          1                      ⁢                      c                                                                                  +                              C                ⁢                                                                  ⁢                                  2                  ·                                      h                                          2                      ⁢                      c                                                                                  +                              C                ⁢                                                                  ⁢                                  3                  ·                                      h                                          3                      ⁢                      c                                                                                  +                                                                                            ⁢                                          C                ⁢                                                                  ⁢                                  4                  ·                                      h                                          4                      ⁢                      c                                                                                  +                              C                ⁢                                                                  ⁢                                  5                  ·                                      h                                          5                      ⁢                      c                                                                                  +                              C                ⁢                                                                  ⁢                                  6                  ·                                      h                                          6                      ⁢                      c                                                                                  +                              2                9                                      ]                    >>          10                                                  d          =                    ⁢                      [                                          A                ⁢                                                                  ⁢                                  3                  ·                                      h                                          1                      ⁢                      d                                                                                  +                              B                ⁢                                                                  ⁢                                  3                  ·                                      h                                          2                      ⁢                      d                                                                                  +                              C                ⁢                                                                  ⁢                                  3                  ·                                      h                                          3                      ⁢                      d                                                                                  +                              D                ⁢                                                                  ⁢                                  3                  ·                                                                                                                          ⁢                                          h                                  4                  ⁢                  d                                            +                              E                ⁢                                                                  ⁢                                  3                  ·                                      h                                          5                      ⁢                      d                                                                                  +                              F                ⁢                                                                  ⁢                                  3                  ·                                      h                                          6                      ⁢                      d                                                                                  +                              2                9                                      ]                    >>          10                                                  l          =                    ⁢                      [                                          A                ⁢                                                                  ⁢                                  3                  ·                                      h                                          1                      ⁢                      l                                                                                  +                              B                ⁢                                                                  ⁢                                  3                  ·                                      h                                          2                      ⁢                      l                                                                                  +                              C                ⁢                                                                  ⁢                                  3                  ·                                      h                                          3                      ⁢                      l                                                                                  +                              D                ⁢                                                                  ⁢                                  3                  ·                                                                                                                          ⁢                                          h                                  4                  ⁢                  l                                            +                              E                ⁢                                                                  ⁢                                  3                  ·                                      h                                          5                      ⁢                      l                                                                                  +                              F                ⁢                                                                  ⁢                                  3                  ·                                      h                                          6                      ⁢                      l                                                                                  +                              2                9                                      ]                    >>          10                                                  h          =                    ⁢                      [                                                            (                                                            A                      ⁢                                                                                          ⁢                      3                                        +                                          F                      ⁢                                                                                          ⁢                      3                                                        )                                ·                                  h                                      1                    ⁢                    h                                                              +                                                (                                                            B                      ⁢                                                                                          ⁢                      3                                        +                                          E                      ⁢                                                                                          ⁢                      3                                                        )                                ·                                                                                                          ⁢                                          h                                  2                  ⁢                  h                                            +                                                (                                                            C                      ⁢                                                                                          ⁢                      3                                        +                                          D                      ⁢                                                                                          ⁢                      3                                                        )                                ·                                  h                                      3                    ⁢                    h                                                              +                              2                9                                      ]                    >>          10                                        f        =                  [                                                                                                                                                                                                                                                                                              (                                                                                                            A                                      ⁢                                                                                                                                                          ⁢                                      1                                                                        +                                                                          A                                      ⁢                                                                                                                                                          ⁢                                      6                                                                                                        )                                                                ·                                                                  h                                                                      11                                    ⁢                                    f                                                                                                                              +                                                                                                (                                                                                                            A                                      ⁢                                                                                                                                                          ⁢                                      2                                                                        +                                                                                                                                                  ⁢                                                                          A                                      ⁢                                                                                                                                                          ⁢                                      5                                                                                                        )                                                                ·                                                                  h                                                                      12                                    ⁢                                    f                                                                                                                              +                                                                                                                                                                                                                                                            (                                                                                                            A                                      ⁢                                                                                                                                                          ⁢                                      3                                                                        +                                                                          A                                      ⁢                                                                                                                                                          ⁢                                      4                                                                                                        )                                                                ·                                                                  h                                                                      13                                    ⁢                                    f                                                                                                                              +                                                                                                (                                                                                                            B                                      ⁢                                                                                                                                                          ⁢                                      1                                                                        +                                                                          B                                      ⁢                                                                                                                                                          ⁢                                      6                                                                                                        )                                                                ·                                                                  h                                                                      21                                    ⁢                                    f                                                                                                                              +                                                                                                                                                                                                                                                                                    (                                                                                          B                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                                                                                          ⁢                                                              B                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          22                              ⁢                              f                                                                                                      +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              B                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          23                              ⁢                              f                                                                                                      +                                                                                                                                                                                                      (                                                                                          C                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              C                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          31                              ⁢                              f                                                                                                      +                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                                                                                          ⁢                                                              C                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          32                              ⁢                              f                                                                                                      +                                                                                                                                                                                                      (                                                                                          C                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              C                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          33                              ⁢                              f                                                                                                      +                                                                              (                                                                                          D                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          41                              ⁢                              f                                                                                                      +                                                                                                                                                                                                      (                                                                                          D                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                                                                                          ⁢                                                              D                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          42                              ⁢                              f                                                                                                      +                                                                              (                                                                                          D                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          43                              ⁢                              f                                                                                                      +                                                                                                                                                                                                      (                                                                                          E                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          51                              ⁢                              f                                                                                                      +                                                                              (                                                                                          E                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                                                                                          ⁢                                                              E                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          52                              ⁢                              f                                                                                                      +                                                                                                                                                                                                      (                                                                                          E                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          53                              ⁢                              f                                                                                                      +                                                                              (                                                                                          F                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              F                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          61                              ⁢                              f                                                                                                      +                                                                                                                                                                                                            (                                                                        F                          ⁢                                                                                                          ⁢                          2                                                +                                                                                                  ⁢                                                  F                          ⁢                                                                                                          ⁢                          5                                                                    )                                        ·                                          h                                              62                        ⁢                        f                                                                              +                                                            (                                                                        F                          ⁢                                                                                                          ⁢                          3                                                +                                                  F                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                                              63                        ⁢                        f                                                                              +                                      2                    9                                                                                ]                    >>      10        ,                  ⁢                  n        =                  [                                                                                                                                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              A                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          11                              ⁢                              n                                                                                                      +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                                                                                          ⁢                                                              A                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          12                              ⁢                              n                                                                                                      +                                                                                                                                                                                                      (                                                                                          A                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              A                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          13                              ⁢                              n                                                                                                      +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              B                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          21                              ⁢                              n                                                                                                      +                                                                                                                                                                                                      (                                                                                          B                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                                                                                          ⁢                                                              B                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          22                              ⁢                              n                                                                                                      +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              B                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          23                              ⁢                              n                                                                                                      +                                                                                                                                                                                                      (                                                                                          C                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              C                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          31                              ⁢                              n                                                                                                      +                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                                                                                          ⁢                                                              C                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          32                              ⁢                              n                                                                                                      +                                                                                                                                                                                                            (                                                                        C                          ⁢                                                                                                          ⁢                          3                                                +                                                  C                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                                              33                        ⁢                        n                                                                              +                                                            (                                                                        D                          ⁢                                                                                                          ⁢                          1                                                +                                                  D                          ⁢                                                                                                          ⁢                          6                                                                    )                                        ·                                          h                                              41                        ⁢                        n                                                                              +                                                                                                                                                (                                                                        D                          ⁢                                                                                                          ⁢                          2                                                +                                                                                                  ⁢                                                  D                          ⁢                                                                                                          ⁢                          5                                                                    )                                        ·                                          h                                              42                        ⁢                        n                                                                              +                                                            (                                                                        D                          ⁢                                                                                                          ⁢                          3                                                +                                                  D                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                                              43                        ⁢                        n                                                                              +                                                                                                                                                                                                                              (                                                                                          E                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          51                              ⁢                              n                                                                                                      +                                                                              (                                                                                          E                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                                                                                          ⁢                                                              E                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          52                              ⁢                              n                                                                                                      +                                                                                                                                                                                                      (                                                                                          E                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          53                              ⁢                              n                                                                                                      +                                                                              (                                                                                          F                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              F                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          61                              ⁢                              n                                                                                                      +                                                                                                                                                                                                            (                                                                        F                          ⁢                                                                                                          ⁢                          2                                                +                                                                                                  ⁢                                                  F                          ⁢                                                                                                          ⁢                          5                                                                    )                                        ·                                          h                                              62                        ⁢                        n                                                                              +                                                            (                                                                        F                          ⁢                                                                                                          ⁢                          3                                                +                                                  F                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                                              63                        ⁢                        n                                                                              +                                      2                    9                                                                                ]                    >>      10        ,                  ⁢                  i        =                  [                                                                                                                                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              F                                ⁢                                                                                                                                  ⁢                                1                                                                                      )                                                    ·                                                      h                                                          11                              ⁢                              i                                                                                                      +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                                                                                          ⁢                                                              F                                ⁢                                                                                                                                  ⁢                                2                                                                                      )                                                    ·                                                      h                                                          12                              ⁢                              i                                                                                                      +                                                                                                                                                                                                      (                                                                                          A                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              F                                ⁢                                                                                                                                  ⁢                                3                                                                                      )                                                    ·                                                      h                                                          13                              ⁢                              i                                                                                                      +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                1                                                                                      )                                                    ·                                                      h                                                          21                              ⁢                              i                                                                                                      +                                                                                                                                                                                                            (                                                                        B                          ⁢                                                                                                          ⁢                          2                                                +                                                                                                  ⁢                                                  E                          ⁢                                                                                                          ⁢                          2                                                                    )                                        ·                                          h                                              22                        ⁢                        i                                                                              +                                                            (                                                                        B                          ⁢                                                                                                          ⁢                          3                                                +                                                  E                          ⁢                                                                                                          ⁢                          3                                                                    )                                        ·                                          h                                              23                        ⁢                        i                                                                              +                                                                                                                                                                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                                                      )                                                    ·                                                      h                                                          31                              ⁢                              i                                                                                                      +                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                                                                                          ⁢                                                              D                                ⁢                                                                                                                                  ⁢                                2                                                                                      )                                                    ·                                                      h                                                          32                              ⁢                              i                                                                                                      +                                                                                                                                                                                                      (                                                                                          C                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                3                                                                                      )                                                    ·                                                      h                                                          33                              ⁢                              i                                                                                                      +                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                6                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          41                              ⁢                              i                                                                                                      +                                                                                                                                                                                                                                                                                          (                                                                                          C                                ⁢                                                                                                                                  ⁢                                5                                                            +                                                                                                                          ⁢                                                              D                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          42                              ⁢                              i                                                                                                      +                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                4                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          43                              ⁢                              i                                                                                                      +                                                                                                                                                                                                      (                                                                                          B                                ⁢                                                                                                                                  ⁢                                6                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          51                              ⁢                              i                                                                                                      +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                5                                                            +                                                                                                                          ⁢                                                              E                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          52                              ⁢                              i                                                                                                      +                                                                                                                                                                                                            (                                                                        B                          ⁢                                                                                                          ⁢                          4                                                +                                                  E                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                                              53                        ⁢                        i                                                                              +                                                            (                                                                        A                          ⁢                                                                                                          ⁢                          6                                                +                                                  F                          ⁢                                                                                                          ⁢                          6                                                                    )                                        ·                                          h                                              61                        ⁢                        i                                                                              +                                                                                                                                                (                                                                        A                          ⁢                                                                                                          ⁢                          5                                                +                                                                                                  ⁢                                                  F                          ⁢                                                                                                          ⁢                          5                                                                    )                                        ·                                          h                                              62                        ⁢                        i                                                                              +                                                            (                                                                        A                          ⁢                                                                                                          ⁢                          4                                                +                                                  F                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                                              63                        ⁢                        i                                                                              +                                      2                    9                                                                                ]                    >>      10        ,                  ⁢                  ⁢                  k        =                  [                                                                                                                                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              F                                ⁢                                                                                                                                  ⁢                                1                                                                                      )                                                    ·                                                      h                                                          11                              ⁢                              k                                                                                                      +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                                                                                          ⁢                                                              F                                ⁢                                                                                                                                  ⁢                                2                                                                                      )                                                    ·                                                      h                                                          12                              ⁢                              k                                                                                                      +                                                                                                                                                                                                      (                                                                                          A                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              F                                ⁢                                                                                                                                  ⁢                                3                                                                                      )                                                    ·                                                      h                                                          13                              ⁢                              k                                                                                                      +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                1                                                                                      )                                                    ·                                                      h                                                          21                              ⁢                              k                                                                                                      +                                                                                                                                                                                                            (                                                                        B                          ⁢                                                                                                          ⁢                          2                                                +                                                                                                  ⁢                                                  E                          ⁢                                                                                                          ⁢                          2                                                                    )                                        ·                                          h                                              22                        ⁢                        k                                                                              +                                                            (                                                                        B                          ⁢                                                                                                          ⁢                          3                                                +                                                  E                          ⁢                                                                                                          ⁢                          3                                                                    )                                        ·                                          h                                              23                        ⁢                        k                                                                              +                                                                                                                                                                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                                                      )                                                    ·                                                      h                                                          31                              ⁢                              k                                                                                                      +                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                                                                                          ⁢                                                              D                                ⁢                                                                                                                                  ⁢                                2                                                                                      )                                                    ·                                                      h                                                          32                              ⁢                              k                                                                                                      +                                                                                                                                                                                                      (                                                                                          C                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                3                                                                                      )                                                    ·                                                      h                                                          33                              ⁢                              k                                                                                                      +                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                6                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          41                              ⁢                              k                                                                                                      +                                                                                                                                                                                                            (                                                                        C                          ⁢                                                                                                          ⁢                          5                                                +                                                                                                  ⁢                                                  D                          ⁢                                                                                                          ⁢                          5                                                                    )                                        ·                                          h                                              42                        ⁢                        k                                                                              +                                                            (                                                                        C                          ⁢                                                                                                          ⁢                          4                                                +                                                  D                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                                              43                        ⁢                        k                                                                              +                                                                                                                                                                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                6                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          51                              ⁢                              k                                                                                                      +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                5                                                            +                                                                                                                          ⁢                                                              E                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          52                              ⁢                              k                                                                                                      +                                                                                                                                                                                                      (                                                                                          B                                ⁢                                                                                                                                  ⁢                                4                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          53                              ⁢                              k                                                                                                      +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                6                                                            +                                                              F                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          61                              ⁢                              k                                                                                                      +                                                                                                                                                                                                            (                                                                        A                          ⁢                                                                                                          ⁢                          5                                                +                                                                                                  ⁢                                                  F                          ⁢                                                                                                          ⁢                          5                                                                    )                                        ·                                          h                                              62                        ⁢                        k                                                                              +                                                            (                                                                        A                          ⁢                                                                                                          ⁢                          4                                                +                                                  F                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                                              63                        ⁢                        k                                                                              +                                      2                    9                                                                                ]                    >>      10        ,                  ⁢                  j        =                  [                                                                                                                (                                                                        A                          ⁢                                                                                                          ⁢                          1                                                +                                                  A                          ⁢                                                                                                          ⁢                          6                                                +                                                  F                          ⁢                                                                                                          ⁢                          1                                                +                                                  F                          ⁢                                                                                                          ⁢                          6                                                                    )                                        ·                                          h                      11                                                        +                                                                                                                                                (                                                                        A                          ⁢                                                                                                          ⁢                          2                                                +                                                  A                          ⁢                                                                                                          ⁢                          5                                                +                                                  B                          ⁢                                                                                                          ⁢                          1                                                +                                                  B                          ⁢                                                                                                          ⁢                          6                                                +                                                  E                          ⁢                                                                                                          ⁢                          1                                                +                                                  E                          ⁢                                                                                                          ⁢                          6                                                +                                                  F                          ⁢                                                                                                          ⁢                          2                                                +                                                  F                          ⁢                                                                                                          ⁢                          5                                                                    )                                        ·                                          h                      12                                                        +                                                                                                                                                (                                                                        A                          ⁢                                                                                                          ⁢                          3                                                +                                                  A                          ⁢                                                                                                          ⁢                          4                                                +                                                  C                          ⁢                                                                                                          ⁢                          1                                                +                                                  C                          ⁢                                                                                                          ⁢                          6                                                +                                                  D                          ⁢                                                                                                          ⁢                          1                                                +                                                  D                          ⁢                                                                                                          ⁢                          6                                                +                                                  F                          ⁢                                                                                                          ⁢                          3                                                +                                                  F                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                      13                                                        +                                                                                                                                                (                                                                        B                          ⁢                                                                                                          ⁢                          2                                                +                                                  B                          ⁢                                                                                                          ⁢                          5                                                +                                                  E                          ⁢                                                                                                          ⁢                          2                                                +                                                  E                          ⁢                                                                                                          ⁢                          5                                                                    )                                        ·                                          h                      22                                                        +                                                                                                                                                (                                                                        B                          ⁢                                                                                                          ⁢                          3                                                +                                                  B                          ⁢                                                                                                          ⁢                          4                                                +                                                  C                          ⁢                                                                                                          ⁢                          2                                                +                                                  C                          ⁢                                                                                                          ⁢                          5                                                +                                                  D                          ⁢                                                                                                          ⁢                          2                                                +                                                  D                          ⁢                                                                                                          ⁢                          5                                                +                                                  E                          ⁢                                                                                                          ⁢                          3                                                +                                                  E                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                      23                                                        +                                                                                                                                                (                                                                        C                          ⁢                                                                                                          ⁢                          3                                                +                                                  C                          ⁢                                                                                                          ⁢                          4                                                +                                                  D                          ⁢                                                                                                          ⁢                          3                                                +                                                  D                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                      33                                                        +                                      2                    9                                                                                ]                    >>      10        ,                  ⁢                  e        =                  [                                                                                                                                                                  A                          ⁢                                                                                                          ⁢                                                      1                            ·                                                          h                                                              11                                ⁢                                e                                                                                                                                    +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                              B                                ⁢                                                                                                                                  ⁢                                1                                                                                      )                                                    ·                                                      h                                                          12                              ⁢                              e                                                                                                      +                                                                                                                                                                                                      (                                                                                          A                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              C                                ⁢                                                                                                                                  ⁢                                1                                                                                      )                                                    ·                                                      h                                                          13                              ⁢                              e                                                                                                      +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                4                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                                                      )                                                    ·                                                      h                                                          14                              ⁢                              e                                                                                                      +                                                                                                                                                                                                            (                                                                        A                          ⁢                                                                                                          ⁢                          5                                                +                                                  E                          ⁢                                                                                                          ⁢                          1                                                                    )                                        ·                                          h                                              15                        ⁢                        e                                                                              +                                                            (                                                                        A                          ⁢                                                                                                          ⁢                          6                                                +                                                  F                          ⁢                                                                                                          ⁢                          1                                                                    )                                        ·                                          h                                              16                        ⁢                        e                                                                              +                                                                                                                                                                                                  B                          ⁢                                                                                                          ⁢                                                      2                            ·                                                          h                                                              22                                ⁢                                e                                                                                                                                    +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              C                                ⁢                                                                                                                                  ⁢                                2                                                                                      )                                                    ·                                                      h                                                          23                              ⁢                              e                                                                                                      +                                                                                                                                                                                                      (                                                                                          B                                ⁢                                                                                                                                  ⁢                                4                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                2                                                                                      )                                                    ·                                                      h                                                          24                              ⁢                              e                                                                                                      +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                5                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                2                                                                                      )                                                    ·                                                      h                                                          25                              ⁢                              e                                                                                                      +                                                                                                                                                                                                            (                                                                        B                          ⁢                                                                                                          ⁢                          6                                                +                                                  F                          ⁢                                                                                                          ⁢                          2                                                                    )                                        ·                                          h                                              26                        ⁢                        e                                                                              +                                      C                    ⁢                                                                                  ⁢                                          3                      ·                                              h                                                  33                          ⁢                          e                                                                                                      +                                                                                                                                                                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                4                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                3                                                                                      )                                                    ·                                                      h                                                          34                              ⁢                              e                                                                                                      +                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                5                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                3                                                                                      )                                                    ·                                                      h                                                          35                              ⁢                              e                                                                                                      +                                                                                                                                                                                                      (                                                                                          C                                ⁢                                                                                                                                  ⁢                                6                                                            +                                                              F                                ⁢                                                                                                                                  ⁢                                3                                                                                      )                                                    ·                                                      h                                                          36                              ⁢                              e                                                                                                      +                                                  D                          ⁢                                                                                                          ⁢                                                      4                            ·                                                          h                                                              44                                ⁢                                e                                                                                                                                    +                                                                                                                                                                                                            (                                                                        D                          ⁢                                                                                                          ⁢                          5                                                +                                                  E                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                                              45                        ⁢                        e                                                                              +                                                            (                                                                        D                          ⁢                                                                                                          ⁢                          6                                                +                                                  F                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                                              46                        ⁢                        e                                                                              +                                                                                                                          E                    ⁢                                                                                  ⁢                                          5                      ·                                              h                                                  55                          ⁢                          e                                                                                                      +                                                            (                                                                        E                          ⁢                                                                                                          ⁢                          6                                                +                                                  F                          ⁢                                                                                                          ⁢                          5                                                                    )                                        ·                                          h                                              56                        ⁢                        e                                                                              +                                      F                    ⁢                                                                                  ⁢                                          6                      ·                                              h                                                  66                          ⁢                          e                                                                                                      +                                      2                    9                                                                                ]                    >>      10        ,                  ⁢                  o        =                  [                                                                                                                                                                  A                          ⁢                                                                                                          ⁢                                                      1                            ·                                                          h                                                              11                                ⁢                                o                                                                                                                                    +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                              B                                ⁢                                                                                                                                  ⁢                                1                                                                                      )                                                    ·                                                      h                                                          12                              ⁢                              o                                                                                                      +                                                                                                                                                                                                      (                                                                                          A                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              C                                ⁢                                                                                                                                  ⁢                                1                                                                                      )                                                    ·                                                      h                                                          13                              ⁢                              o                                                                                                      +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                4                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                                                      )                                                    ·                                                      h                                                          14                              ⁢                              o                                                                                                      +                                                                                                                                                                                                            (                                                                        A                          ⁢                                                                                                          ⁢                          5                                                +                                                  E                          ⁢                                                                                                          ⁢                          1                                                                    )                                        ·                                          h                                              15                        ⁢                        o                                                                              +                                                            (                                                                        A                          ⁢                                                                                                          ⁢                          6                                                +                                                  F                          ⁢                                                                                                          ⁢                          1                                                                    )                                        ·                                          h                                              16                        ⁢                        o                                                                              +                                                                                                                                                                                                  B                          ⁢                                                                                                          ⁢                                                      2                            ·                                                          h                                                              22                                ⁢                                o                                                                                                                                    +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              C                                ⁢                                                                                                                                  ⁢                                2                                                                                      )                                                    ·                                                      h                                                          23                              ⁢                              o                                                                                                      +                                                                                                                                                                                                      (                                                                                          B                                ⁢                                                                                                                                  ⁢                                4                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                2                                                                                      )                                                    ·                                                      h                                                          24                              ⁢                              o                                                                                                      +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                5                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                2                                                                                      )                                                    ·                                                      h                                                          25                              ⁢                              o                                                                                                      +                                                                                                                                                                                                            (                                                                        B                          ⁢                                                                                                          ⁢                          6                                                +                                                  F                          ⁢                                                                                                          ⁢                          2                                                                    )                                        ·                                          h                                              26                        ⁢                        o                                                                              +                                      C                    ⁢                                                                                  ⁢                                          3                      ·                                              h                                                  33                          ⁢                          o                                                                                                      +                                                                                                                                                                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                4                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                3                                                                                      )                                                    ·                                                      h                                                          34                              ⁢                              o                                                                                                      +                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                5                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                3                                                                                      )                                                    ·                                                      h                                                          35                              ⁢                              o                                                                                                      +                                                                                                                                                                                                      (                                                                                          C                                ⁢                                                                                                                                  ⁢                                6                                                            +                                                              F                                ⁢                                                                                                                                  ⁢                                3                                                                                      )                                                    ·                                                      h                                                          36                              ⁢                              o                                                                                                      +                                                  D                          ⁢                                                                                                          ⁢                                                      4                            ·                                                          h                                                              44                                ⁢                                o                                                                                                                                    +                                                                                                                                                                                                            (                                                                        D                          ⁢                                                                                                          ⁢                          5                                                +                                                  E                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                                              45                        ⁢                        o                                                                              +                                                            (                                                                        D                          ⁢                                                                                                          ⁢                          6                                                +                                                  F                          ⁢                                                                                                          ⁢                          4                                                                    )                                        ·                                          h                                              46                        ⁢                        o                                                                              +                                                                                                                          E                    ⁢                                                                                  ⁢                                          5                      ·                                              h                                                  55                          ⁢                          o                                                                                                      +                                                            (                                                                        E                          ⁢                                                                                                          ⁢                          6                                                +                                                  F                          ⁢                                                                                                          ⁢                          5                                                                    )                                        ·                                          h                                              56                        ⁢                        o                                                                              +                                      F                    ⁢                                                                                  ⁢                                          6                      ·                                              h                                                  66                          ⁢                          o                                                                                                      +                                      2                    9                                                                                ]                    >>      10        ,                  ⁢                  g        =                  [                                                                                                                                                                  A                          ⁢                                                                                                          ⁢                                                      6                            ·                                                          h                                                              11                                ⁢                                g                                                                                                                                    +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                5                                                            +                                                              B                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          12                              ⁢                              g                                                                                                      +                                                                                                                                                                                                      (                                                                                          A                                ⁢                                                                                                                                  ⁢                                4                                                            +                                                              C                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          13                              ⁢                              g                                                                                                      +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          14                              ⁢                              g                                                                                                      +                                                                                                                                                                                                            (                                                                        A                          ⁢                                                                                                          ⁢                          2                                                +                                                  E                          ⁢                                                                                                          ⁢                          6                                                                    )                                        ·                                          h                                              15                        ⁢                        g                                                                              +                                                            (                                                                        A                          ⁢                                                                                                          ⁢                          1                                                +                                                  F                          ⁢                                                                                                          ⁢                          6                                                                    )                                        ·                                          h                                              16                        ⁢                        g                                                                              +                                                                                                                                                                                                  B                          ⁢                                                                                                          ⁢                                                      5                            ·                                                          h                                                              22                                ⁢                                g                                                                                                                                    +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                4                                                            +                                                              C                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          23                              ⁢                              g                                                                                                      +                                                                                                                                                                                                      (                                                                                          B                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          24                              ⁢                              g                                                                                                      +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          25                              ⁢                              g                                                                                                      +                                                                                                                                                                                                            (                                                                        B                          ⁢                                                                                                          ⁢                          1                                                +                                                  F                          ⁢                                                                                                          ⁢                          5                                                                    )                                        ·                                          h                                              26                        ⁢                        g                                                                              +                                      C                    ⁢                                                                                  ⁢                                          4                      ·                                              h                                                  33                          ⁢                          g                                                                                                      +                                                                                                                                                                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          34                              ⁢                              g                                                                                                      +                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          35                              ⁢                              g                                                                                                      +                                                                                                                                                                                                      (                                                                                          C                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              F                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          36                              ⁢                              g                                                                                                      +                                                  D                          ⁢                                                                                                          ⁢                                                      3                            ·                                                          h                                                              44                                ⁢                                g                                                                                                                                    +                                                                                                                                                                                                            (                                                                        D                          ⁢                                                                                                          ⁢                          2                                                +                                                  E                          ⁢                                                                                                          ⁢                          3                                                                    )                                        ·                                          h                                              45                        ⁢                        g                                                                              +                                                            (                                                                        D                          ⁢                                                                                                          ⁢                          1                                                +                                                  F                          ⁢                                                                                                          ⁢                          3                                                                    )                                        ·                                          h                                              46                        ⁢                        g                                                                              +                                                                                                                          E                    ⁢                                                                                  ⁢                                          2                      ·                                              h                                                  55                          ⁢                          g                                                                                                      +                                                            (                                                                        E                          ⁢                                                                                                          ⁢                          1                                                +                                                  F                          ⁢                                                                                                          ⁢                          2                                                                    )                                        ·                                          h                                              56                        ⁢                        g                                                                              +                                      F                    ⁢                                                                                  ⁢                                          1                      ·                                              h                                                  66                          ⁢                          g                                                                                                      +                                      2                    9                                                                                ]                    >>      10        ,                  ⁢                  m        =                  [                                                                                                                                                                  A                          ⁢                                                                                                          ⁢                                                      6                            ·                                                          h                                                              11                                ⁢                                m                                                                                                                                    +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                5                                                            +                                                              B                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          12                              ⁢                              m                                                                                                      +                                                                                                                                                                                                      (                                                                                          A                                ⁢                                                                                                                                  ⁢                                4                                                            +                                                              C                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          13                              ⁢                              m                                                                                                      +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                6                                                                                      )                                                    ·                                                      h                                                          14                              ⁢                              m                                                                                                      +                                                                                                                                                                                                            (                                                                        A                          ⁢                                                                                                          ⁢                          2                                                +                                                  E                          ⁢                                                                                                          ⁢                          6                                                                    )                                        ·                                          h                                              15                        ⁢                        m                                                                              +                                                            (                                                                        A                          ⁢                                                                                                          ⁢                          1                                                +                                                  F                          ⁢                                                                                                          ⁢                          6                                                                    )                                        ·                                          h                                              16                        ⁢                        m                                                                              +                                                                                                                                                                                                  B                          ⁢                                                                                                          ⁢                                                      5                            ·                                                          h                                                              22                                ⁢                                m                                                                                                                                    +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                4                                                            +                                                              C                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          23                              ⁢                              m                                                                                                      +                                                                                                                                                                                                      (                                                                                          B                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          24                              ⁢                              m                                                                                                      +                                                                              (                                                                                          B                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                5                                                                                      )                                                    ·                                                      h                                                          25                              ⁢                              m                                                                                                      +                                                                                                                                                                                                            (                                                                        B                          ⁢                                                                                                          ⁢                          1                                                +                                                  F                          ⁢                                                                                                          ⁢                          5                                                                    )                                        ·                                          h                                              26                        ⁢                        m                                                                              +                                      C                    ⁢                                                                                  ⁢                                          4                      ·                                              h                                                  33                          ⁢                          m                                                                                                      +                                                                                                                                                                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                3                                                            +                                                              D                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          34                              ⁢                              m                                                                                                      +                                                                              (                                                                                          C                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          35                              ⁢                              m                                                                                                      +                                                                                                                                                                                                      (                                                                                          C                                ⁢                                                                                                                                  ⁢                                1                                                            +                                                              F                                ⁢                                                                                                                                  ⁢                                4                                                                                      )                                                    ·                                                      h                                                          36                              ⁢                              m                                                                                                      +                                                  D                          ⁢                                                                                                          ⁢                                                      3                            ·                                                          h                                                              44                                ⁢                                m                                                                                                                                    +                                                                              (                                                                                          D                                ⁢                                                                                                                                  ⁢                                2                                                            +                                                              E                                ⁢                                                                                                                                  ⁢                                3                                                                                      )                                                    ·                                                      h                                                          45                              ⁢                              m                                                                                                      +                                                                                                                                                                                                            (                                                                        D                          ⁢                                                                                                          ⁢                          1                                                +                                                  F                          ⁢                                                                                                          ⁢                          3                                                                    )                                        ·                                          h                                              46                        ⁢                        m                                                                              +                                      E                    ⁢                                                                                  ⁢                                          2                      ·                                              h                                                  55                          ⁢                          m                                                                                                      +                                                                                                                                                (                                                                        E                          ⁢                                                                                                          ⁢                          1                                                +                                                  F                          ⁢                                                                                                          ⁢                          2                                                                    )                                        ·                                          h                                              56                        ⁢                        m                                                                              +                                      F                    ⁢                                                                                  ⁢                                          1                      ·                                              h                                                  66                          ⁢                          m                                                                                                      +                                      2                    9                                                                                ]                    >>      10        ,  
hxyz indicates the interpolation coefficient needed by each point during the interpolation.
For this two-dimensional interpolation filter, all the sub-pixel points are obtained by performing linear weighting on the integer pixel points. a, b, c, d, h, and l adopt the one-dimensional interpolation filter method. Other points undergo the interpolation filter by using 36 integer pixel points in the two-dimensional space.
The two-dimensional interpolation filter greatly increases the complexity of the encoder and the decoder. In addition, when small-size images are encoded, a lot of integer pixel point information is also needed. Thus, there is a lot of interpolation coefficient information to be transmitted in the bit stream, increasing the burden of the bit stream and reducing the efficiency of transmitting the interpolation coefficients by the bit stream.
Solution 3: In the current VCEG solution, a frame-level adaptive one-dimensional interpolation filter method is available. This method simplifies the interpolation operation on the basis of the preceding two-dimensional interpolation filter. In this method, the interpolation is performed with several one-dimensional horizontal and one-dimensional vertical integer pixel points only. Details are as follows:a=[C1·h1a+C2·h2a+C3·h3a+C4·h4a+C5·h5a+C6·h6a+27]>>8b=[(C1+C6)·h1b+(C2+C5)·h2b+(C3+C4)·h3b+27]>>8c=[C1·h1c+C2·h2c+C3·h3c+C4·h4c+C5·h5c+C6·h6c+27]>>8h=[(A3+F3)·h1h+(B3+E3)·h2h+(C3+D3)·h3h+27]>>8j=[(bA+bF)·h1j+(bB+bE)·h2j+(bC+bD)·h3j+27]>>8d=[A3·h1d+B3·h2d+C3·h3d+D3·h4d+E3·h5d+F3·h6d+27]>>8l=[A3·h1l+B3·h2l+C3·h3l+D3·h4l+E3·h5l+F3·h6l+27]>>8i=[(aA+aF)·h1i+(aB+aE)·h2i+(aC+aD)·h3i+27]>>8k=[(cA+cB)·h1k+(cB+cE)·h2k+(cC+cD)·h3k+27]>>8f=[bA·h1f+bB·h2f+bC·h3f+bD·h4f+bE·h5f+bF·h6f+27]>>8n=[bA·h1n+bB·h2n+bC·h3n+bD·h4n+bE·h5n+bF·h6n+27]>>8e=[(aA·h1e+aB·h2e+aC·h3e+aD·h4e+aE·h5e+aF·h6e+27]>>8m=[(aA·h1m+aB·h2m+aC·h3m+aD·h4m+aE·h5m+aF·h6m+27]>>8g=[(cA·h1g+cB·h2g+cC·h3g+cD·h4g+cE·h5g+cF·h6g+27]>>8o=[(cA·h1o+cB·h2o+cC·h3o+cD·h4o+cE·h5o+cF·h6o+27]>>8
hxy indicates the interpolation coefficient needed by each point during the interpolation.
Compared with the preceding two-dimensional interpolation filter, the one-dimensional interpolation filter simplifies the interpolation operation. This interpolation method, however, cannot be fully compatible with the prior art (H.264). In this method, some integer pixel points or sub-pixel points need to be extended to perform the interpolation operation. In addition, sub-pixel points a, b, and c undergo a one-dimensional interpolation by using the horizontal pixel information, while sub-pixel points d, e, f, g, h, I, j, k, l, m, n, and o undergo a one-dimensional interpolation by using the vertical pixel information. In this separable one-dimensional interpolation filter method, too many vertical pixel points of the image are used. This may reduce the encoding efficiency when the texture information of the vertical pixels of the image changes frequently.